Topological edge states with ultracold atoms carrying orbital angular momentum in a diamond chain

We study the single-particle properties of a system formed by ultracold atoms loaded into the manifold of $l=1$ orbital angular momentum (OAM) states of an optical lattice with a diamond-chain geometry. Through a series of successive basis rotations, we show that the OAM degree of freedom induces phases in some tunneling amplitudes of the tight-binding model that are equivalent to a net $\pi$ flux through the plaquettes. These effects give rise to a topologically nontrivial band structure and protected edge states which persist everywhere in the parameter space of the model, indicating the absence of a topological transition. By taking advantage of these analytical mappings, we also show that this system constitutes a realization of a square-root topological insulator. In addition, we demonstrate that quantum interferences between the different tunneling processes involved in the dynamics may lead to Aharanov-Bohm caging in the system. All these analytical results are confirmed by exact diagonalization numerical calculations.

Figure 1

Sketch of the diamond-chain optical lattice considered in this work, indicating the shape of the unit cell as well as the two OAM $l=1$ states that can be occupied on each site and their associated operators. The directions along which all the couplings are real are signaled with double-arrow solid blue lines, whereas the double-arrow dashed red lines are drawn in the directions along which the tunneling couplings involving a change in the circulation, i.e., those whose amplitude is $J_1$ or $J_3$ in (5), acquire a $\pi$ phase.

Figure 2

Dependence of the $J_1$, $J_2$, and $J_3$ tunneling amplitudes on the trap separation $d$ for two ring potentials of radius (a) $R=0$ (harmonic traps), (b) $R=2.5\sigma$, and (c) $R=5.0\sigma$. In each of the plots, the insets show the values of the tunneling amplitudes for large values of $d$, where $J_2\approx J_3$ and $|J_1|\ll |J_2|,|J_3|$.

Figure 3

Energy bands of the diamond chain in the OAM $l=1$ manifold computed using the values of $J_2$ and $J_3$ that are obtained for harmonic potentials separated by distances $d=3.5\sigma$ (a) and $d=6\sigma$ (b), shown in Fig. 2.

Figure 4

Sketch of the two decoupled diamond chains $H^{+}$ (a) and $H^{-}$ (b) that are obtained after performing the basis rotation (10).

Figure 5

(a) Sketch of the modified SSH chain that is obtained after performing the rotation (13) in the $H^{+}$ diamond chain that was obtained after the first basis rotation (10). (b) Modified SSH chain in the ${\mathrm{\Omega }}_3=0$ ($J_2=J_3$) limit, where the bulk sites become decoupled in trimers and an isolated dimer appears at the right edge.

Figure 6

Numerically computed time evolution, for a system of $N_c=5$ unit cells, of the population of the states $|A_i,+\rangle$ (black solid lines) and $|A_i,-\rangle$ (blue dotted lines) and of the total sum of the populations of the states $|B_3,\pm \rangle ,|C_3,\pm \rangle ,|A_3,\pm \rangle ,|B_4,\pm \rangle ,|C_4,\pm \rangle$ (red dashed-dotted lines). The tunneling parameters are (a) $J_3=1.1J_2$ and (b) $J_3=J_2$. In both cases, the initial state is $|\mathrm{\Psi }\rangle =\sqrt{0.7}|A_i,+\rangle +\sqrt{0.3}|A_i,-\rangle$.

Figure 7

Schematic representation of the mapping from the modified SSH $H^{+}$ chain into a diamond chain with alternate hoppings. The two possible choices for the inversion-symmetry axis within a given unit cell, under periodic boundary conditions, are also shown. Note that neither of them are localized at the central axis of the unit cell.

Figure 8

Sketch of the tight-binding model obtained after squaring the bulk Hamiltonian of the modified diamond-chain model (25).

Figure 9

(a) Band structure of the squared model (26). (b) Squared energy spectrum of an open diamond with $N_c=40$ unit cells in the original OAM $l=1$ model. In both plots, the parameters are $t_2=0.2t_1$.

Figure 10

Energy spectra of the single-particle diamond-chain Hamiltonian (5) obtained with the values of $J_2$ and $J_3$ corresponding to harmonic traps separated by distances $d=3.5\sigma$, $4.5\sigma$, and $6.0\sigma$ (in h.o. units). The number of unit cells considered is $N_c=20$.

Figure 11

Sum of the populations of all the states localized on the $A$ sites for the different eigenstates of the single-particle diamond-chain Hamiltonian (5), for $J_2=J_3$ and $N_c=20$ unit cells.

Figure 12

Density profiles of one of the in-gap states (spectral index $i=39$) of a diamond chain with a total number of unit cells $N_c=20$ computed for different relative values of $J_2$ and $J_3$. The upper plot corresponds to the density distribution of the states with negative circulation and the lower plot to the states of positive circulation.

Figure 13

(a) Schematic representation of the left end of the modified SSH chain in the presence of a nonzero value of $J_1$ for $J_2\ne J_3$ (left) and $J_1$ for $J_2=J_3$ (right). In the latter case, the onsite potential $V$ shifts the energies of four states with respect to the case $J_1=0$. (b) Energy spectrum of a diamond lattice with 20 unit cells and tunneling parameters $J_2=J_3=-10J_1$. A total of eight states, which are indicated on the plot by blue circles, have small shifts with respect to the case $J_1=0$.